ODE
\[ x^2 y'(x)=y(x) \left (a x+b y(x)^3\right ) \] ODE Classification
[[_homogeneous, `class G`], _rational, _Bernoulli]
Book solution method
The Bernoulli ODE
Mathematica ✓
cpu = 0.031254 (sec), leaf count = 144
\[\left \{\left \{y(x)\to \frac {\sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}}\right \},\left \{y(x)\to \frac {(-1)^{2/3} \sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}}\right \}\right \}\]
Maple ✓
cpu = 0.008 (sec), leaf count = 29
\[ \left \{ 3\,{\frac {b}{x \left ( 3\,a-1 \right ) }}-{x}^{-3\,a}{\it \_C1}+ \left ( y \left ( x \right ) \right ) ^{-3}=0 \right \} \] Mathematica raw input
DSolve[x^2*y'[x] == y[x]*(a*x + b*y[x]^3),y[x],x]
Mathematica raw output
{{y[x] -> ((1 - 3*a)*x^(1 + 3*a))^(1/3)/(3*b*x^(3*a) + (1 - 3*a)*x*C[1])^(1/3)}, {y[x] -> -(((-1)^(1/3)*((1 - 3*a)*x^(1 + 3*a))^(1/3))/(3*b*x^(3*a) + (1 - 3*a)*x*C[1])^(1/3))}, {y[x] -> ((-1)^(2/3)*((1 - 3*a)*x^(1 + 3*a))^(1/3))/(3*b*x^(3*a) + (1 - 3*a)*x*C[1])^(1/3)}}
Maple raw input
dsolve(x^2*diff(y(x),x) = (a*x+b*y(x)^3)*y(x), y(x),'implicit')
Maple raw output
3/x/(3*a-1)*b-x^(-3*a)*_C1+1/y(x)^3 = 0